| 释义 |
Diophantine Equation--QuadraticAn equation of the form
 | (1) |
 is an Integer is called a Pell Equation. Pell equations, as well as the analogous equation with a minussign on the right, can be solved by finding the Continued Fraction for  . (The trivial solution  , is ignored in all subsequent discussion.) Let  denote the  th Convergent  ,then we are looking for a convergent which obeys the identity
 | (2) |
 , where  , i.e.,
 | (3) |
If  is Odd, then  is Positive and the solution in terms of smallest Integers is  and  , where  is the th Convergent. If is Even, then is Negative, but
 | (4) |
 ,  . Summarizing,
 | (5) |
The more complicated equation
 | (6) |
 and , but the procedure is more complicated (Chrystal 1961). However, ifa single solution to the above equation is known, other solutions can be found. Let  and  be solutions to(6), and and  solutions to the ``unit'' form. Then
 | (7) |
 | (8) |
Call a Diophantine equation consisting of finding  Powers equal to a sum of equalPowers an `` equation.'' The 2-1 equation
 | (9) |
 ,  ,  ) has a well-known general solution (Dickson 1966,pp. 165-170). To solve the equation, note that every Prime of the form  can be expressed as the sum of twoRelatively Prime squares in exactly one way. To find in how many ways a general number can be expressed as a sumof two squares, factor it as follows
 | (10) |
 and the s are primes of the form  . If the  s are integral, thendefine
 | (11) |
 | (12) |
If zero is counted as a square, both Positive and Negative numbers are included, and the order of the two squares isdistinguished, Jacobi  showed that the number of ways a number can be written as the sum of two squares is four times theexcess of the number of Divisors of the form over the number of Divisors ofthe form .
A set of Integers satisfying the 3-1 equation
 | (13) |
 | (14) |
Solutions to an equation of the form
 | (15) |
 | (16) |
 | (17) |
Ramanujan's Square Equation
 | (19) |
 , 4, 5, 7, and 15 (Beeler et al. 1972, Item 31).See also Algebra, Cannonball Problem, Continued Fraction, Fermat Difference Equation,Lagrange Number (Diophantine Equation), Pell Equation, Pythagorean Quadruple, PythagoreanTriple, Quadratic Residue
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.Beiler, A. H. ``The Pellian.'' Ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, pp. 248-268, 1966. Bell, E. T. The Development of Mathematics, 2nd ed. New York: McGraw-Hill, p. 159, 1945. Chrystal, G. Textbook of Algebra, 2 vols. New York: Chelsea, 1961. Degan, C. F. Canon Pellianus. Copenhagen, Denmark, 1817. Dickson, L. E. ``Number of Representations as a Sum of 5, 6, 7, or 8 Squares.'' Ch. 13 in Studies in the Theory of Numbers. Chicago, IL: University of Chicago Press, 1930. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1966. Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994. Lam, T. Y. The Algebraic Theory of Quadratic Forms. Reading, MA: W. A. Benjamin, 1973. Rajwade, A. R. Squares. Cambridge, England: Cambridge University Press, 1993. Scharlau, W. Quadratic and Hermitian Forms. Berlin: Springer-Verlag, 1985. Shapiro, D. B. ``Products of Sums and Squares.'' Expo. Math. 2, 235-261, 1984. Smarandache, F. ``Un metodo de resolucion de la ecuacion diofantica.'' Gaz. Math. 1, 151-157, 1988. Smarandache, F. ``Method to Solve the Diophantine Equation  .'' In Collected Papers, Vol. 1. Bucharest, Romania: Tempus, 1996. Taussky, O. ``Sums of Squares.'' Amer. Math. Monthly 77, 805-830, 1970. Whitford, E. E. Pell Equation. New York: Columbia University Press, 1912.
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